strongly interacting matter and light far from equilibrium · strongly interacting matter and light...

Strongly Interacting Matter and Light Far From Equilibrium

Marco Schiro’ CNRS-IPhT Saclay

Center For Optical Quantum Technologies, Hamburg April 2015

Condensed Matter

Ultra-Cold Atoms

Quantum Many Body Physics with “Light” (?)

Light used to probe phases or (more recently) to ‘’induce’’

phases of matter

Light used to trap (optical lattices) to probe or to excite atoms

Non-Linear Optics with Single Photons at Finite Density

Bloch group, Munich

Cavalleri group H

amburg

Houck group, Princeton

Outline

Coupling Light and Matter at the Quantum Level

Many Body Physics with “Atoms” and “Photons”: Perspectives and Challenges

Quantum Phase Transitions of Light in Arrays Coupled CQED Units

Exploring Light-Matter Interaction at

the Quantum Level

How Strong Can the Coupling Between Light and Matter Be?

M. Devoret, S. Girvin&R. Schoelkopf, Ann Phys 16 767 (2007)

g = dat E0/~

Fine Structure Constant Limit

g

Coupling via current (i.e. magnetic field) can be even larger!

dat eL~r

4 0

2E2

0 V

g r

L

r

r2

‘’Poor-Man’’ Estimate of Light-Matter Dipole Coupling

g eL

rr

2 ~0 V

Cavity Quantum ElectroDynamics

Trapping photons and atoms into cavities!

Light-Matter coupling (Dipole)

J. M. Raimond, M. Brune and S. Haroche, Rev. Mod. Phys. 73, 565 (2001)

Hint d ·E

Alkali or Rydberg Atoms trapped into the cavity

Optical/Microwave 3D cavities supporting discrete modes

A Solid-State Analogue: Circuit QED

Microwave Transmission Line Resonators

Circuit QED

A. Blais et al, Phys. Rev. A 69 062320 (2004)

Artificial Atoms: Superconducting Qubits

Superconducting-based to avoid dissipation

Mesoscopic SC system with ‘‘atom-like’‘ spectrum

Capacitive/Inductive Coupling

Energy Scales in CQEDCircuit QEDCavity QED

Photon Losses and Atomic Decay Rates (Dissipation)

g , Strong-Coupling Regime of cQED

Photon Frequency/Atom splitting r,q

Light-Matter Coupling (Vacuum Rabi Splitting)

,

g r,q

Typical Circuit QED:

g 1Ghz , 500 kHz!r 10Ghz

Effective Photon-Photon Interactions

by Light-Matter Coupling

Quantum Models of Light and Matter

I. Rabi (1936)

Harmonic Mode

(Photon)

Anharmonic Mode (Atom)

Linear coupling (dipole)

Circuit QEDCavity QED

Often in Rotating Wave Approximation (requires )g min(r,q)

Jaynes,Cumming (1963)

X

HRabi = r a†a+ q

+ + ga† + a

+ +

HRabi = r a†a+ q

+ + ga† + hc

+ g

a†+ + hc

HJC = r a

†a+ q+ + g

a† + hc

The Rabi Model

The Jaynes-Cumming Model

Photon-Blockaded Transport

A.J Hoffmann et al, Phys. Rev. Lett 107, (2011)

HJC r a†a+ Ueff

a†a

2g |r q|

Ueff g4/|r q|3

Strongly Dispersive Regime:

Effective Photon Hamiltonian

Steps in the Photon Transmission through a driven cavity

Many Body Physics with Strongly Interacting Light and

Matter

Exciton-Polariton Condensate Cold-Atoms in Optical Cavities

H.Ritsch, P. Domokos, F. Brennecke, T. Esslinger, RMP 85 (2013)I. Carusotto, C. Ciuti, RMP 85 (2013)

Driven-Dissipative Superfluid Self-Organization/Dicke Transition

Exp: ETH (Esslinger), Hamburg (Hemmerich)Exp: Paris (J.Bloch), Stanford (Yamamoto),..

Coupled Arrays of CQED UnitsExperiments at Princeton: (@ A. Houck’s Lab)

Hopping between adjacent resonators/cavities

Effective Photon-Photon Interactions

Delocalized bosonic particles with an effective onsite interaction...

Bose-Hubbard Physics(?)

HJCL =X

ij

Jij a†i aj +

X

i

HiJC

Jaynes-Cumming Lattice Model

J. Koch and K. Le Hur PRA 80 023811 (2009)

The importance of being a Photon

µph =E

N= 0

Unlike massive particles Photons are not conserved in Nature

Photons have zero chemical potential!

The density of a Photon Gas is set by drive-dissipation or interaction with matter!

Think of QED!

Can we drive photons into steady states with non trivial

quantum correlations?

Two possible directions

Balancing Drive and DissipationCritical Behavior for Driven-Dissipative Quantum Systems?

Exotic Non-Equilibrium Steady States?

External Drive? Coherent vs Incoherent?

Increase Light-Matter Coupling: effective driving Counter-Rotating Terms

g r,q

Hint = HJC + ga†+ + a

‘Ultra-Strong’ Coupling Regime

Photon Circuit QED Lattices with Rabi Non Linearity

M. Schiro, M. Bordyuh, B. Oztop and H. Tureci, Phys. Rev. Lett. 109, 053601 (2012)

M. Schiro, M. Bordyuh, B. Oztop and H. Tureci, J. Phys. B: At. Mol. Opt. Phys. 46 224021(2013)

Ultra-Strong Coupling of Light and Matter

Ultra-Strong Coupling Regime with Superconducting Circuits

Devoret et al (2010)

T. Niemczyk et al, Nat Phys (2010)

Inductively Coupled Flux Qubit

g/r ' 0.12

Is the ‘‘standard’’ CQED picture working?S. Nigg et al, Phys. Rev. Lett 108 240502 (2012)

Many Other Platforms: Quantum Dots, 2DEG,WG Superlattices,..Todorov et al, PRL (2010), G. Scalari et al Science (2012), A.Crespi et al PRL (2012),..

Deviations from JC Physics

P. Nataf & C. Ciuti, Nat. Comm (2010), O. Viehnmann et al, PRL (2011)

No Chemical Potential!

Light-Matter interaction fix the average number of excitations in the ground state

Can we have a non-trivial ground-state due to an ‘‘effective’’ drive?

HiRabi = r a

†iai + q

+i

i + g

a†i + ai

+i +

i

The Rabi Lattice Model

HRL =X

ij

Jij a†i aj +

X

i

HiRabi

Insights from the Rabi Model

Z2 symmetry enough for an ‘‘exact’’ solution!

D. Braak, Phys Rev Lett 107 100401 (2011)

Symmetries: Parity † a = a

† x = x

= eiNpolZ2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

g/tr

-1

0

1

2

3

4

< N >< a >< a2 >

Finite Number of Excitations, Squeezed, Symmetric

0 2 4 6 8 10n0

0.1

0.2

0.3

0.4

0.5

P n- = | c

n- |2

g = 1.5 tr

Non Trivial Ground-state Properties = !r !q = 0

|Rabi =X

n,=±cn |n

HRabi = r a†a+ q

+ + ga† + a

+ +

A Low Energy Doublet at Strong Coupling

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

g/tr-1

-0.5

0

0.5

1

1.5

2

Energy

W = + 1

W = < 1

W = + 1

W = < 1

Low-energy spectrum

First excited state, with opposite parity, is almost degenerate for

= !r !q = 0

Ground-state has always the same (even) parity

g r

= r e2(g/r)

2

Exponentially Small Splitting!

HRabi = r a†a+ q

+ + ga† + a

+ +

Small Hopping: Disordered Phase, Gapped, Insulating

Large Hopping: Ordered Phase, Gapped

h aii 6= 0hx

i

i 6= 0

hx

i

i = 0 h aii = 0

Qualitative Phase Diagram in the J vs g plane:

What happens at finite (small) Hopping?

HRL =X

ij

Jij a†i aj +

X

i

HiRabi

Z2 Parity Symmetry Breaking

Quantum Phases and QPT are different from the physics of massive quantum particles on a lattice!

Equilibrium Phase Diagram

‘’Para-electric’’ Insulator

h aii = 0

hx

i

i = 0

h aii 6= 0

0 0.2 0.4 0.6 0.8

J/ω0

0.5

1

1.5

2g

/ω0

δ = − 0.5δ = 0.0δ = 0.5δ = 1.0δ = 2.0

Negative detuning favors the ordering phase

= !q !r

Ising Universality

Z2 Parity Symmetry Breaking

‘‘Ferro-electric’’ Insulator

hx

i

i 6= 0

Tunable Open Rabi Hubbard Model with Engineered Driving

M. Schiro, C. Joshi, M. Bordyuh, R. Fazio, J. Keeling and H. Tureci, arXiv: 1503.04456

Driven 4-level Atom in a CavityH0 = !cav a

†a+X

j=0,1,r,s

Ej | jih j|

Hg = gr | rih 0| a+ gs | sih 1| a+ hc

Hdrive(t) =

r

2ei!p

r t | rih 1|+ s

2ei!p

s t | sih 0|+ hc

Cavity Photon mediate excited state transition

Two-pump scheme driving, Raman assisted transitions

For large detunings eliminate excited states and get an effective Hamiltonian for the 0,1 manifold

|0ih1| |1ih0| +

r/s gr/s,r/s,!pr/s

F. Dimer et al, PRA (2007) M. Baden et al, PRL (2014)

“Rotating-Frame”: time-independent Hamiltonian in a non-equilibrium, Markovian Bath

Tunable Light-Matter Couplings

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

1.1

g/ω

R

0 0.2 0.4 0.6 0.8 1

Ωs/Ω

r

0

0.2

0.4

0.6

0.8

1

g’/

g

Heff = !R a†a+ !qz + g

a† + hc

+ g0

a†+ + hc

g =grr

2r

g0 =gss

2s

0 0.2 0.4 0.6 0.8 1

Ωs/Ω

r

-8

-6

-4

-2

0

δ

α = 0.010α = 0.012α = 0.013α = 0.014

Parameters of the effective model in the regime

= !q !R

g/!R g/g0 1

Steady State Patterns? Broken Symmetry? Open Quantum Criticality?

Driven-Dissipative Cavity Array

κ κ κ κ

Pump

PumpCavity

Cavity

HRL =X

ij

Jij a†i aj +

X

i

HiRabi

@t = i[HRL, ] + L[, ] + L[a, ]Dissipative vs Unitary Dynamics

L[J, ] = 2OJ† J†J J†J Jumps Operator

(t) =Y

i

i(t) Time-Dependent Gutzwiller Ansatz

Steady State Phase Diagram (I)

g

0

0.5

1

1.5

2

0 π/2 π

(a)Most unstable k

g

⟨σx

nσx

n+1⟩

J=0.45J=0.9

0

0.5

1

1.5

2-0.5 0 0.5

(b) ⟨σx

nσx

n+1⟩ vs g

⟨σx nσx n+

1⟩

J

g=1.5g=0.5-0.5

0

0.5

0 0.25 0.5 0.75

(c) ⟨σx

nσx

n+1⟩ vs J

⟨σx nσx n+l⟩

l

J=0.4J=0.6

J=0.8

0 4 8 12

-0.5

0

0.5(d) ⟨σ

x

nσx

n+l⟩vs l, g=1.5

g = g0

Commensurate-to-Incommensurate Order as J,g varied

Critical Hopping Jc pushed to finite value in the open case

Steady State Phase Diagram (II)

g

J

0

1

2 0 0.25 0.5 0.75

(a) Most unstable kg'/g=0.5

J

0 0.25 0.5 0.75

0

π/2

π(b) Most unstable k

g'/g=0.25⟨σ

x nσ

x n+l⟩

g

l=1l=2

l=3l=4

-0.2

0

0.2

0 0.6 1.2 1.8

(c) ⟨σxnσ

xn+l⟩ vs g

l

g=0.16g=0.32

g=0.64g=0.80

0 4 8

(d) ⟨σxnσ

xn+l⟩ vs l

g'/g=0.25, J=0.5

g 6= g0

Upon decreasing of g’/g: suppression of order, then transition to AF ordering

0 2 4 6 8 10

r-10

-5

0

5

J eff(r

,ω)

ω = 0.0ω = 0.5 ω

0

ω = 2ω0

ω = 4 ω0

Photon-Induced Qbit InteractionsPhotons always appear quadratically

and can be ‘‘traced’’ out

Effective exchange interaction between quits

Exchange depends from photon band-structure/

population

Short-distance component from “ferro” to “anti-ferro” as frequency increases

Sg = g

Zdt

X

i

a†i (t)

i (t) + hc

Sint

=

Zdtdt0

X

ij

x

i

(t)Jeff

ij

(t t0)x

j

(t0)

Jeffij (!) = g2

X

k

cosk (Ri Rj)! !k

(! !k)2+ 2

From Rabi single site: low-energy doublet with a tiny splitting

Universal Behavior at the Rabi QPT

Opposite Parity: ‘‘Isospin’’

x|± = ±|±

Anisotropic XY model ! Quantum Ising Universality Class

Heff

=X

h iji

Jx x

i

x

j

+ Jy y

i

y

j

+

2

X

i

z

i

Project onto this low energy doublet:

Rabi Hamiltonian in this subspace

HiRabi !

2z

i

“Renormalized” Dissipation D[] = +L[+, ] + L[, ]

Ferro-Antiferro-Ferro Transition

Suppression of ordering (Ferro/Normal/Antiferro): Energy Splitting

vanishes at gc

Antiferro Z2 ordering coincides with population inversion!

Infinite-MPO Simulation of Effective Dissipative Spin Model

confirms the picture!

Summing UpDriven-Dissipative Light Matter Systems

Promising Interface Quantum Optics/Condensed Matter

Perspectives:

• Real Circuit QED architecture (flux qubits?) • Effect of Disorder

Tunable Open Rabi Hubbard Array by Engineered Driving

Platform for non-equilibrium quantum many body physics

• Coupling strongly correlated electrons to cavity fields?Y. Laplace, S. Fernandez-Pena, S. Gariglio, J.M. Triscone, A. Cavalleri, arXiv:1503.06117

Acknowledgements

In collaboration with:

Discussions with:

Andrew Houck, D.Underwood, D. Sadri, D. Huse (Princeton) Takis Kontos(ENS), Nicolas Roch (Grenoble), K. Le Hur(Polytechnique)

Hakan Tureci, Mykola Bordyuh, Baris Oztop (Princeton)

Chatanya Joshi, Jonathan Keeling (St. Andrews)

Saro Fazio (Pisa)

Thanks!

Interplay of Drive and DissipationTime-Dependent Drive can be “often” gauged away by

going to a “rotating-frame”

(t) = †(t)(t)(t)

H(t) = i†(t)@t(t) †(t)H(t)(t)

@t = i[H(t), (t)]

In this rotating frame Hamiltonian becomes time-independent..

…but occupation of bath modes become non-equilibrium!

GKbath(!) = ImGR

bath(!) coth

! + drive

2

Violation of Fluctuation-Dissipation Theorem

J=0 -- Polaritons

J<Jc -- Mott Insulator of Polaritons

J>Jc -- ‘’Superfluid’’

Hartmann et al, Greentree et al, Schmidt et al,Rossini et al, Tomadin et al,..

[HJC

, Npol

] = 0

haii 6= 0

U(1) Spontaneous Symmetry Breaking J. Koch and K. Le Hur PRA 80 023811 (2009)

Mott to Superfluid Transition of Polaritons

HJCL

=X

ij

Jij

a†i

aj

+ h.c.+

X

i

Hi

JC

µX

i

N i

pol

Jaynes-Cumming Lattice Model

The Role of Counter-Rotating Terms: From Jaynes-Cumming to Rabi

M. Schiro et al, (in preparation)

From Jaynes-Cumming to Rabi

Structure of the Ground State vs g0/g

Pn = | cn|2 Probabilty of having n polaritons| =

X

n,=±cn |n

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Pn-

g’/g = 0

0 2 4 6 8 100

0.2

0.4

0.6

0.8g’/g = 0.25

0 2 4 6 8 10n0

0.1

0.2

0.3

0.4

Pn-

g’/g = 1.0

0 2 4 6 8 10n0

0.1

0.2

0.3

0.4

0.5g’/g = 0.75

0 0.2 0.4 0.6 0.8 1

g’/g

0

0.5

1

1.5

2

2.5

3

3.5

4

g/ω

0

Π = +1

Π = −1

Π = +1

Π = −1

Level Crossings between different parity sectors at finite g0/g

HRabi = r a† a+ q

+ + g (a† + hc) + g(a†+ + hc)

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

Jc/ω

0

g = 0.5 ω0

0 0.25 0.5 0.75 10

0.05

0.1

Jc/ω

0

g = 1.1 ω0

0 0.25 0.5 0.75 1

g’/g

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

Jc/ω

0

g = 3.5 ω0

0 0.25 0.5 0.75 1

g’/g

1e-06

0.0001

0.01

Jc/ω

0

g = 2.5 ω0

Π = +1

Π = +1

Π = −1

Π = −1

Π = +1Π = −1

< a > = 0 < σx >=0

Π = +1Π = −1

Π = +1

The Fate of Mott Lobes

Lobes at fixed Parity survive at any finite but shift upg0/g 6= 1

What happens to the Jaynes-Cumming Mott Lobes as we tune g’/g?

The Role of Counter-Rotating Terms

Effective Action Z = Tr eH =

Zi Di D

i eSeff [i ,i]

Seff [i ,i] =

Z

0d

X

h iji

i J

1ij j +

X

i

[i ,i]

“Local” Physics (Rabi Non-Linearity)

M. Fisher et al, PRB(1989) for Bose-Hubbard

Standard Field Theory of U(1) Superfluid-to-Mott Transition...

.....+ Relevant Operators (explicit breaking down to Z2)

K. Damle & S. Sachdev, PRL(96)

[,] = logh eR 0 d(() a()+a†()())i

Seff =

Z

0d

K1

+K2||2 +K3||2 + r||2 + u||4

SCRT =

Z

0d g0 (+)

Basics of Superconducting Circuits (I)

Transmission Line Resonator

Circuit ‘‘Quantization’’

Hr =X

n

n a†n an

n = n

dplc

Superconducting-based to give dissipation-less currents

Hr =

Z d

0dx

q2(x)

2c+

2(x)

2l

q(x) =X

n

An(x)an + a

†n

(x) = iX

n

Bn(x)an a†n

M. Devoret, Les Houches (1995)

Qubit: Mesoscopic SC system with ‘‘atom-like’’ spectrum

The Josephson Junction is the crucial (non-linear) ingredient!

Qubits come in different flavors (charge, flux, phase qubits)

Basics of Superconducting Circuits (II)

Ex:The Cooper Pair Box

HCPB = 4EC (n ng)2 EJ cos

Ultra-Strong Coupling of Light and Matter

Ultra-Strong Coupling Regime with Superconducting Circuits

Devoret et al (2010)

How large can be the coupling between atom and photon in circuit QED?

M. Devoret, S. Girvin and R. Schoelkopf,

Ann Phys 16 767 (2007)

T. Niemczyk et al, Nat Phys (2010)

Inductively Coupled Flux Qubit

g/r ' 0.12

Is the ‘‘standard’’ CQED picture working?S. Nigg et al, Phys. Rev. Lett 108 240502 (2012)

Many Other Platforms: Quantum Dots, 2DEG,WG Superlattices,..Todorov et al, PRL (2010), G. Scalari et al Science (2012), A.Crespi et al PRL (2012),..

Deviations from JC Physics

P. Nataf & C. Ciuti, Nat. Comm (2010), O. Viehnmann et al, PRL (2011)

Disordered Arrays of Coupled Cavities

D. Underwood, W. Shanks, J. Koch and A. Houck, PRA 86, 023837 (2012)

H =X

i

(r + i) a†i ai +

X

ij

tija†i aj + hc

tij = 2Z0 Cij (r + i) (r + j)

strongly interacting matter and light far from equilibrium · strongly interacting matter and light...

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